Mean square of the Hurwitz zeta-function and other remarks

International audience The Hurwitz zeta-function associated with the parameter $a\,(0< a\leq1)$ is a generalisation of the Riemann zeta-function namely the case $a=1$. It is defined by $$\zeta(s,a)=\sum_{n=0}^{\infty}(n+a)^{-s},\,(s=\sigma+it,\,\sigma>1)$$ and its analytic continuation. %In fact $$\zeta(s,a)=\sum_{n=0}^{\infty}\left((n+a)^{-s}-\int_{n}^{n+1}\frac{du}{(u+a)^s} \right)+\frac{a^{1-s}}{s-1}$$ gives the analytic continuation to $(\sigma>0)$. A repetition of this several times shows that $$\zeta-\frac{a^{1-s}}{s-1}$$ can be continued as an entire function to the whole plane. In $Re(s)\geq-1,\,t\geq2,\,\zeta(s,a)-a^{-s}=O(t^3)$ and by the functional equation (see \S2) it is $$O\left(\left(\frac{\vert s\vert}{2\pi}\right)^{\frac{1}{2}-Re(s)}\right)$$ in $Re(s)\leq-1,\,t\geq2$. From these facts In this paper, we deduce an `Approximate function equation' (see \S3), which is a generalisation of the approximate functional equation for $\zeta(s)$. Combining this with an important theorem due to van-der-Corput, we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\zeta(\frac{1}{2}+it)-a^{-\frac{1}{2}-it}\vert^2 dt <\!\!\!< (\log T)^3$$ uniformly in $a(0< a\leq1)$. From this we deduce similar results for quasi $L$-functions and more general functions. %Let $a_1, a_2,\ldots$, be any periodic sequence of complex numbers for which the sum over a period is zero. Let $b_1, b_2,\ldots$ be any sequence of complex numbers for which $\sum_{j=2}^{n}\vert b_j-b_{j-1}\vert+\vert b_n\vert\leq n^{\varepsilon}$ for every $\varepsilon>0$ and every $n\geq n_0(\varepsilon)$. Then we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\sum_{n=1}^{\infty}\frac{a_nb_n}{(n+a)^{\frac{1}{2}+it}}\vert^2\,dt\leq T^{\varepsilon}$$ for every $\varepsilon>0$ and every $T\geq T_0(\varepsilon)$. Here, as usual, $0<a\leq1$ and $T_0(\varepsilon)$ is independent of $a$.